A note on the upper bound and girth pair of (k;g)(k;g)-cages

A (k;g)(k;g)-cage   is a kk-regular graph of girth gg with minimum order. In this work, for all k≥3k≥3 and g≥5g≥5 odd, we present an upper bound of the order of a (k;g+1)(k;g+1)-cage in terms of the order of a (k;g)(k;g)-cage, improving a previous result by Sauer of 1967. We also show that every (k;11)(k;11)-cage with k≥6k≥6 contains a cycle of length 12, supporting a conjecture by Harary and Kovács of 1983.